舍盖·索伯列夫Sergei Sobolev

舍盖·索伯列夫的父亲Lev Aleksandrovich 索伯列夫是一位重要的律师和大律师。他的母亲Nataliya Georgievna在索伯列夫的成长中发挥了重要作用，尤其是在索伯列夫的父亲去世之后，当时索伯列夫14岁。他在哈尔科夫工人技术学校学习，准备进入中学，并于1922年在他父亲去世前后进入了中学。

他进入的那所中学当时称为列宁格勒第190学校，尽管此前它被称为连托夫斯基中学。在5中（另见4）解释说，这所学校：-

……是在第一次俄国革命期间，由圣彼得堡最杰出的教师们为那些因参与革命运动而被国家学校和技术学院开除的学生创办的。

1925年中学毕业后，索伯列夫进入列宁格勒国立大学物理数学系，他的才能很快被三年前回到列宁格勒的弗拉迪米尔·斯米尔诺夫发现。索伯列夫对微分方程产生了兴趣，这个主题将贯穿他一生的研究，甚至在他职业生涯的这个阶段，他就产生了新的成果并予以发表。

到1929年，索伯列夫完成了大学教育，并开始在一些不同的教育机构任教。例如，他的第一个职位是1929年在苏联科学院地震研究所理论部。然而，此外，他还在1930-31年在列宁格勒电工学院任教。

1932年，弗拉基米尔·安德烈耶维奇·斯捷克洛夫物理数学研究所分为数学系和物理系。伊万·维诺格拉多夫领导数学系，并邀请索伯列夫加入该系。然而此时，索伯列夫已经（[4]和[5]）：-

……发表了一系列深刻的论文，在其中他提出了一种求解重要一类偏微分方程的新方法。

与弗拉迪米尔·斯米尔诺夫合作，索伯列夫研究了波动方程的泛函不变解。这些方法使他们能够找到描述弹性介质振荡的波动方程的闭式解。这些方法还引导他们建立了完整的瑞利表面波理论，而索伯列夫继续解决了衍射问题。索伯列夫因这项杰出工作于1933年当选为苏联科学院通讯院士，获得荣誉。

1934年4月28日，在苏联科学院数学与自然科学部的全体会议上，决定将两年前创建的弗拉基米尔·安德烈耶维奇·斯捷克洛夫物理数学研究所的各部拆分为两个独立研究所，即弗拉基米尔·安德烈耶维奇·斯捷克洛夫数学研究所和列别捷夫物理研究所。同年，弗拉基米尔·安德烈耶维奇·斯捷克洛夫数学研究所从列宁格勒迁往莫斯科，索伯列夫随新研究所前往莫斯科。到1935年，索伯列夫是该研究所微分方程理论部主任。

在20世纪30年代，索伯列夫引入了若干概念，这些概念对数学中几个不同领域的发展具有根本性的意义[22]：-

对索伯列夫函数空间的研究，他在20世纪30年代引入了这些空间，随即成为泛函分析的一个完整领域。索伯列夫的广义函数（分布）概念显得尤为重要；随着洛朗·施瓦茨和伊斯拉埃尔·盖尔范德的进一步发展，它成为数学的核心概念之一。

在莫斯科工作期间，索伯列夫在求解椭圆型边值问题的标准变分方法基础上，通过引入这些索伯列夫函数空间进行了发展。他给出了这些空间上范数的不等式，这些不等式在嵌入函数空间理论中很重要。他将自己的方法应用于解决数学物理中的困难问题。

1939年，索伯列夫当选为苏联科学院正式成员。他当选时年仅31岁，这是一项非凡的成就。这使他成为科学院最年轻的正式成员，事实上他在相当长的一段时间内一直是最年轻的成员。

第二次世界大战开始时，弗拉基米尔·安德烈耶维奇·斯捷克洛夫数学研究所从莫斯科迁至喀山。同年10月，索伯列夫被任命为该研究所所长，并于1943年春季监督研究所迁回莫斯科。索伯列夫在1941年首次颁发斯大林奖（后称国家奖）时成为首批获奖者之一。他担任研究所所长的时期于1944年2月结束。

他研究的一个新领域涉及旋转容器中流体的运动。他被引导去研究若干新问题（[4]和[5]）：-

……使他奠定了不定度规空间中算子理论的基础，并在算子谱理论中引入了新思想。这些思想主要涉及非经典边值问题的广义解。

1950年，他以俄文出版了他的著名著作Applications of functional analysis in mathematical physics。1963年，American Mathematical Society出版了英译本。同年，洛朗·施瓦茨关于广义函数论的书也问世了。

20世纪50年代初，索伯列夫的工作转向计算数学，并于1952年成为苏联第一个计算数学系的负责人，当时他在莫斯科国立大学组织了第一个这样的系。然而在1956年，他与一些同事一起提出了通过教育举措开发俄罗斯东部大片地区的方案。该计划是建立若干科学研究所，以平衡苏联东部大量高质量教育机构。

计划获批后，索伯列夫在莫斯科待了一段时间，招募人员并组织在新西伯利亚建立研究所。在[4]和[5]中，强调了他对新西伯利亚研究所的贡献：-

在研究所艰难的初创岁月里，索伯列夫以自身的卓越榜样，将科研工作的最佳习惯灌输给年轻同事。十年间，在索伯列夫的领导下，USSR Academy of Sciences西伯利亚分院数学研究所已成为具有国际地位的最伟大的数学科学中心之一。

1958年，索伯列夫是苏联代表团成员，参加国际数学联盟，代表团由伊万·维诺格拉多夫率领；同年，索伯列夫出席了爱丁堡国际大会，并作了关于偏微分方程的邀请报告。从1960年到1978年，索伯列夫除了在苏联科学院西伯利亚分院数学研究所工作外，还是新西伯利亚大学的教授。

在20世纪60年代，索伯列夫的大部分研究都指向数值方法，特别是插值法。虽然单变量函数的插值已经研究得很透彻，但多维插值问题在很大程度上仍未解决。索伯列夫将他的广义函数理论和函数空间嵌入理论应用于求积公式，即单变量函数求积公式的多维类似物。同样，索伯列夫的Introduction to the theory of cubature formulae这部重要著作在这一领域极具影响力。

索伯列夫因其对数学的根本性贡献而获得了许多荣誉。他当选为许多科学学会的成员，包括苏联科学院、Académie des Sciences de France和Accademia Nazionale dei Lincei。他获得了许多奖项，包括三项国家奖和1988年由苏联科学院颁发的M V罗蒙诺索夫金质奖章。

Sergei L'vovich Sobolev's father, Lev Aleksandrovich Sobolev, was an important layer and barrister. His mother, Nataliya Georgievna, played an important role in Sobolev's upbringing, particularly after the death of Sobolev's father when Sobolev was 14 years old. He studied at the Khar'kov Workers' Technical School preparing to enter the high school which he did in 1922 around the time of his father's death.

The high school which he entered was called the 190th School of Leningrad at the time although previously it had been called the Lentovskii High School. In [5] (see also [4]) it is explained that this school :-

... was founded during the First Russian Revolution by the foremost St Petersburg teachers for pupils who had been excluded from the State Schools and Technical Colleges because of their participation in the revolutionary movement.

After graduating from high school in 1925, Sobolev entered the Physics and Mathematics Faculty of Leningrad State University where his talents were quickly spotted by Smirnov who had returned to Leningrad three years earlier. Sobolev became interested in differential equations, a topic which would dominate his research throughout his life, and even at this stage in his career he produced new results which he published.

By 1929 Sobolev had completed his university education and he began to teach in a number of different educational establishments. For example his first appointment was in 1929 at the Theoretical Department of the Seismological Institute of the USSR Academy of Sciences. However, in addition, he taught at the Leningrad Electrotechnic Institute in 1930-31.

In 1932 the Steklov Institute of Physics and Mathematics was divided into separate Departments of Mathematics and of Physics. Vinogradov headed the Mathematics Department and invited Sobolev to join the Department. By this time, however, Sobolev had already ([4] and [5]):-

... published a number of profound papers in which he put forward a new method for the solution of an important class of partial differential equations.

Working with Smirnov, Sobolev studied functionally invariant solutions of the wave equation. These methods allowed them to find closed form solutions to the wave equation describing the oscillations of an elastic medium. The methods also led them to a complete theory of Rayleigh surface waves and Sobolev went on to solve problems on diffraction. Sobolev was honoured for this outstanding work by election as Corresponding Member of the USSR Academy of Sciences in 1933.

On 28 April 1934, at the general meeting of the Division of Mathematical and Natural Sciences of the USSR Academy of Sciences, a decision was taken to split the Departments of the Steklov Institute of Physics and Mathematics, which had been created two years earlier, into two independent Institutes, the Steklov Mathematical Institute and the Lebedev Physical Institute. In the same year the Steklov Mathematical Institute was moved from Leningrad to Moscow and Sobolev went with the new Institute to Moscow. By 1935 Sobolev was head of the Department of the Theory of Differential Equations at the Institute.

During the 1930s Sobolev introduced notions which were fundamental in the development of several different areas of mathematics [22]:-

The study of Sobolev function spaces, which he introduced in the 1930s, immediately became a whole area of functional analysis. Sobolev's notion of generalised function (distribution) turned out to be especially important; with further developments by Schwartz and Gelfand, it became one of the central notions of mathematics.

While working in Moscow, Sobolev built on the standard variational method for solving elliptic boundary value problems by introducing these Sobolev function spaces. He gave inequalities on the norms on these spaces which were important in the theory of embedding function spaces. He applied his methods to solve difficult problems in mathematical physics.

In 1939 Sobolev was elected a full member of the USSR Academy of Sciences. He was only 31 years of age at the time of his election which was a remarkable achievement. It made him the youngest full member of the Academy of Sciences and in fact he remained the youngest member for quite a few years.

At the beginning of World War II, the Steklov Mathematical Institute was moved from Moscow to Kazan. In the October of that year Sobolev was appointed as Director of the Institute and in the spring of 1943 he supervised the move of the Institute back to Moscow. Sobolev became one of the first recipients of a Stalin prize (later called a State prize) in the first presentation of these prizes in 1941. His period as Director of the Institute ended in February 1944.

A new area of his research involved the study of the motion of a fluid in a rotating vessel. He was led to study a number of new problems which ([4] and [5]):-

... led him to lay the foundations of the theory of operators in a space with an indefinite metric, and to introduce new ideas in the spectral theory of operators. These ideas in the main concern generalised solutions of non-classical boundary value problems.

In 1950 he published his famous text Applications of functional analysis in mathematical physics (in Russian). An English translation was published by the American Mathematical Society in 1963. Schwartz's book on the the theory of distributions appeared in the same year.

In the early 1950s Sobolev's work turned towards computational mathematics and in 1952 he became head of the first department of computational mathematics in the Soviet Union when he organised the first such department at Moscow State University. However in 1956 he joined with a number of colleagues in proposing ways in which the large areas of Russia in the east could be opened up with educational initiatives. The scheme was to set up a number of Institutes for Scientific research to balance the large number of high quality educational establishments in the east of the Soviet Union.

After the plan was approved, Sobolev spent some time in Moscow recruiting staff and organising the establishment of an Institute in Novosibirsk. In [4] and [5] his contribution to the Institute in Novosibirsk is stressed:-

During the difficult formative years of the Institute Sobolev, by his excellent example, infused his young colleagues with the best habits for scientific work. In ten years, under the leadership of Sobolev, the Institute of Mathematics of the Siberian Branch of the USSR Academy of Sciences has become one of the greatest centres for the mathematical sciences of international status.

In 1958 Sobolev was part of the Soviet delegation to the International Mathematical Union, the delegation being led by Vinogradov, and Sobolev attended the International Congress at Edinburgh that year and gave an invited address on partial differential equations. From 1960 until 1978 Sobolev, in addition to his work at the Institute of Mathematics of the Siberian Branch of the USSR Academy of Sciences, was a professor at Novosibirsk University.

During the 1960s much of Sobolev's research was directed towards numerical methods, in particular to interpolation. Although interpolation for functions of a single variable was well worked out, the problem of interpolation in many dimensions was largely unsolved. Sobolev applied his theories of generalised functions and of embeddings of function spaces to cubature formulae, the multi-dimensional analogues of quadrature formulae for functions of one variable. Again a major text by Sobolev Introduction to the theory of cubature formulae has been extremely influential in this area.

Sobolev received many honours for his fundamental contributions to mathematics. He was elected to many scientific societies, including the USSR Academy of Sciences, the Académie des Sciences de France, and the Accademia Nazionale dei Lincei. He was awarded many prizes, including three State Prizes and the 1988 M V Lomonosov Gold Medal from the USSR Academy of Sciences.